Joural of Mathematcal Aalyss ad Applcatos 245, 502 52 2000 do:0.006 jmaa.2000.6774, avalable ole at http: www.dealbrary.com o The Bru Mkowsk Iequalty, Mkowsk s Frst Iequalty, ad Ther Duals R. J. Garder Departmet of Mathematcs, Wester Washgto U ersty, Bellgham, Washgto 98225-9063 E-mal: garder@baker.math.wwu.edu ad S. Vassallo U ersta ` cattolca del S. Cuore, Largo Gemell, I-2023 Mla, Italy E-mal: vassallo@m.ucatt.t Submtted by Wllam F. Ames Receved August 4, 999 Quattatve versos are gve of the equvalece of the Bru Mkowsk equalty ad Mkowsk s frst equalty from the Bru Mkowsk theory. Smlar quattatve versos are obtaed of the equvalece of the correspodg equaltes from Lutwak s dual Bru Mkowsk theory. The ma results are show to be the best possble up to costat factors. 2000 Academc Press Key Words: geometrc tomography; star body; covex body; Bru Mkowsk equalty; Mkowsk s frst equalty; mxed volume; dual Bru Mkowsk equalty; dual mxed volume.. INTRODUCTION Let K ad L be covex bodes. The Bru Mkowsk equalty states that V Ž t. K tl Ž t. VŽ K. tv Ž L., The frst author s supported part by NSF Grat DMS 9802388. The secod author s supported part by Itala Research Coucl CNR Grat 96.00226.CT0. 0022-247X 00 $35.00 Copyrght 2000 by Academc Press All rghts of reproducto ay form reserved. 502
BRUNN MINKOWSKI INEQUALITY 503 where VŽ K. deotes the volume of K, stads for Mkowsk or vector addto, ad 0 t. Ž Secto 2 explas other terms ad otato.. The equalty s a corerstoe of the extesve Bru Mkowsk theory, expouded Scheder s book 5. Secto 6. of ths book provdes proofs ad may hstorcal remarks. Its power geometry s llustrated by the fact that the sopermetrc equalty s a very specal case; see 6, pp. 368 372. The Bru Mkowsk equalty holds whe K, L, ad Ž t. K tl are oempty bouded measurable sets ad bears o aalyss va several extesos that may be formulated as equaltes betwee tegrals of measurable fuctos, as s explaed 2. Varous applcatos have surfaced other areas of mathematcs, for example, to probablty ad multvarate statstcs, shapes of crystals, geometrc tomography, ellptc partal dfferetal equatos, ad combatorcs; see, 3, 6, 2, 5, Secto 6., 6. Usg the homogeety property of volume, the Bru Mkowsk equalty ca be wrtte the smpler but equvalet form VŽ K L. 2 0, Ž. where K ad L are covex bodes wth VŽ K. VŽ L.. It s kow to be equvalet to Mkowsk s Ž frst. equalty V K, L 0, 2 where K ad L are covex bodes wth VŽ K. VŽ L. ; see 6, Theorem B.2., 5, Secto 6.2. Here V Ž K, L. deotes the mxed volume of K ad L whch K appears Ž. tmes ad L oce. Mkowsk s frst equalty plays a role the soluto of Shephard s problem o projectos of cetrally symmetrc covex bodes; see 6, Chap. 4, 4, p. 255. Let M ad N be star bodes wth VŽ M. VŽ N.. The dual Bru Mkowsk equalty 6, Ž B.28., p. 374, 3 states that 2 VŽ M N. 0, Ž 3. where deotes radal addto. It s equvalet to the dual Mkowsk equalty 6, Ž B.24., p. 373, 3 V M, N 0. 4 These equaltes, dual to Ž. ad Ž. 2, respectvely, are fudametal tools a dual Bru Mkowsk theory, tated by Lutwak 3 ad developed by hm ad others. There s a dual sopermetrc equalty that s a specal case of Ž.Ž 3 see 6, p. 373.. The dual Mkowsk equalty plays a part
504 GARDNER AND VASSALLO the soluto of the Busema Petty problem o cetral sectos of cetered covex bodes gve 4, 5, 7, 8. The assumpto cocerg the volume of the bodes above s smply a matter of coveece. Geeral versos of these equaltes ca easly be obtaed from them usg the homogeety property of volume. The authors 7, 8 recetly bega a study of stroger forms, sometmes called stablty versos, of such equaltes the dual Bru Mkowsk theory, spred by smlar studes Žsee Groemer s survey. the Bru Mkowsk theory that go back to Mkowsk hmself. A cosequece of the results 8 s the relatoshp betwee the left-had sdes of Ž. 3 ad Ž. 4, ž / 2 VŽ M N. V Ž M, N. 2Ž. 2 VŽ M N., Ž 5. where M ad N are star bodes wth VŽ M. VŽ N.. Of course Ž. 5 mples the kow fact that Ž. 3 ad Ž. 4 are equvalet, but the quattatve lk provded by Ž. 5 s much more formatve. Moreover, t was show 8 that Ž. 5 s the best possble up to costat factors. Here we gve a drect proof of the more dffcult rght-had equalty Ž. 5. I fact, Theorem 3.2 below provdes a slght mprovemet whch the factor 2 s replaced by 2 Ž2.. Aother motvato for the ew proof s that, ulke the proof 8, t s easly modfed to yeld a correspodg result the Bru Mkowsk theory, stated the tro- ducto 8, that reads as follows. Let K ad L be covex bodes such that VŽ K. VŽ L.. There s a 0 such that f V Ž K, L., the V K L 2 V K, L V K L 2, 6 for ay 2 Ž2.. The left-had equalty was establshed earler by Groemer 0, p. 2 Žsee also 5, p. 39. ; t does ot requre the extra assumpto V Ž K, L., whch, however, s essetal for the rght-had equalty Ž see Theorem 4. ad the paragraph followg t.. I Theorem 3.3 below we use the same method to show that where VŽ M, N. Ž 2 VŽ M N..,. Ž 2.
BRUNN MINKOWSKI INEQUALITY 505 Aga, ths s the best possble up to a costat factor, as Example 3.4 demostrates. As we prove Theorem 4.2, ths verso has a more satsfyg couterpart the Bru Mkowsk theory, amely, VŽ K, L. Ž VŽ K L. 2.. Example 4.3 below shows the factor caot be mproved. Therefore the prevous equalty ad the left-had equalty of Ž. 6 together provde a best possble quattatve expresso of the equvalece of the Bru Mkowsk equalty ad Mkowsk s frst equalty. 2. DEFINITIONS AND PRELIMINARIES We deote the org, ut sphere, ad closed ut ball -dmesoal Eucldea space by o, S, ad B, respectvely. Lebesgue k-dmesoal measure k, k,...,, ca be det- fed wth k-dmesoal Hausdorff measure. The sphercal Lebesgue measure S ca be detfed wth S. I ths paper tegrato over S wth respect to wll be deoted by du. We wrte V, ad call ths olume. We also wrte VŽ B.. We say that a set s cetered f t s cetrally symmetrc, wth ceter at the org. A co ex body s a compact covex set wth oempty teror. Let K ad L be covex bodes. Mkowsk s theorem o mxed volumes Žsee 5, Theorem 5..6. mples that Ý ž / VŽ K L. VŽ K, ; L,.. Ž 7. The followg equalty holds whe 0 j k Žsee 5, Ž 6.4.5., p. 334. : k k j j j k V K, L V K, L V K, L. 8 Here VŽ K, L. deotes the mxed volume of K ad L whch K appears Ž. tmes ad L appears tmes. Note that V K, L V K ad V K, L V L. 0
506 GARDNER AND VASSALLO There s a useful formula for V K, L : VŽ K L. VŽ K. VŽ K, L. lm. Ž 9. 0 The quattes VŽ K, L. ca be expressed as averages of volumes of projectos o subspaces. The reader s referred to 5 for a comprehesve accout of the Bru Mkowsk theory. A set M s star-shaped at the org f every le through the org that meets M does so a Ž possbly degeerate. closed le segmet. If M s a set that s star-shaped at the org, ts radal fucto M s defed, for all u S such that the le through the org parallel to u tersects M, by M Ž u. maxc : cu M 4. I ths paper, a star body s a set that s star-shaped at the org ad whose radal fucto s postve ad cotuous o S. There are other deftos of ths term the lterature; see, for example, 9. If x, y, the the radal sum x y of x ad y s defed to be the usual vector sum x y f x ad y are cotaed a le through o, ad o otherwse. If M ad N are star bodes ad s, t, the sm tn sx ty : x M, y N 4 ad sm tn s M t N. I 3, Lutwak defed the dual mxed olume V Ž M,...,M. of star bodes M,...,M by He observed that ṼŽ M,...,M. Ž u. Ž u. Ž u. du. H M M2 M S Ý ž / VŽ M N. V Ž M, ; N,.. Ž 0. See, for example, 6, Theorem A.6.. We ca also cosstetly defe for star bodes M ad N ad, H M N S V M, N V M, ; N, u u du.
BRUNN MINKOWSKI INEQUALITY 507 Note that V Ž M, N. VŽ M. ad V Ž M, N. VŽ N.. 0 The quattes V Ž M, N. ca be expressed as averages of volumes of sectos by subspaces. Lutwak 3 Žsee also 6, Ž B.22., p. 373. proved that whe, j, ad k are real umbers satsfyg j k, V M, N V M, N V M, N, k k j j j k wth equalty f ad oly f M s a dlatate of N. LEMMA 3.. 3. THE DUAL BRUNN MINKOWSKI AND DUAL MINKOWSKI INEQUALITIES If 2 ad 0 x, the x Ž 2 Ž x. x.. Ž 2. Proof. Let fž x. Ž 2 Ž x. x. x. Ž 2. Ž. Ž. Note that f 0 2 0 ad f 0. Also, But 2 f Ž x. Ž Ž x. x Ž 2. x.. Ž 2. 2 2 2 2 Ý Ž x. x x x Ž 2. x, 2 sce x whe 0 x. So f Ž x. 0 ad fž x. 0 for 0 x. THEOREM 3.2. Let M ad N be star bodes such that VŽ M. V N ad let 2 Ž2.The. ž / V Ž M, N. Ž. 2 VŽ M N..
508 Proof. GARDNER AND VASSALLO Takg ad k Ž., we see that Ž j. Ž. V Ž M, N. V Ž M, N., j for j. Usg ths ad 0, we obta VŽ M N. Ý V Ž M, N. Ý V Ž M, N. Ž. Ž. Ž. V M, N V M, N. Ž. Wth x V M, N, we have to show that for 0 x. Let x Ž. ž 2 Ž Ž x. x. /, 2 x x gž x., 2 ad ote that 0 g x whe 0 x. The by Lemma 3.. ž / Ž. 2 Ž x. x Ž. 2 Ž. gž x. 2 Ž. gž x. 2 Ž x. x Ž 2. x, THEOREM 3.3. Let M ad N be star bodes such that VŽ M. VŽ N. ad let. Ž 2.
BRUNN MINKOWSKI INEQUALITY 509 The VŽ M, N. Ž 2 VŽ M N... Proof. Followg the frst part of the proof of Theorem 3.2, ad aga Ž. lettg x V M, N, we are led to precsely the equalty of Lemma 3.. EXAMPLE 3.4. The followg example s a specal case of 8, Example 5.2. Let M B be the cetered ball wth VŽ M., ad let N be a closed half-ball wth ceter at the org ad ut volume. The Ž u. N 2 whe u s a closed half of S, ad zero otherwse. Note that N s ot a star body because N s ot cotuous, but we ca approxmate N arbtrarly closely by star bodes. By drect computato we obta ad Ž. VŽ M, N. 2 f Ž 2. ž 2 / 2 VŽ M N. 2 gž., say. It was oted 8, Example 5.2 that f ŽŽ. g. decreases to Ž 2l2. as, showg that Theorem 3.2 s the best possble up to the costat factor Ž 2l2.. Let Ž 2. 2 VŽ M N. 2 gž., ž 2 / say. It ca be show that f Ž g. decreases to Ž 4 2 2. as, so Theorem 3.3 s also best possble up to ths costat factor. ' 4. THE BRUNN MINKOWSKI INEQUALITY AND MINKOWSKI S FIRST INEQUALITY THEOREM 4.. Let K ad L be co ex bodes such that VŽ K. V L, ad let 2 Ž2.There. s a 0 such that f V Ž K, L., the Ž. V K, L V K L 2. 2
50 Proof. GARDNER AND VASSALLO Takg ad k Ž. 8, we see that j V Ž K, L. V Ž K, L., j Ž. for j. Usg ths ad 7, we obta VŽ K L. Ý V Ž K, L. Ý VŽ K, L. Ž. Ž. Ž. V K, L V K, L. Ž. Wth x V K, L, we have to show that there s a 0 such that for x. Now ad x ž Ž x x. 2, / 3 Ž x. 2 x Ž Ž x. x. 2 ž 2 / Ý Ž x. 2 x Ž x. Ž x. 2 x Ž x. AŽ x., say, where Ax Ž2. as x. Therefore the rght-had sde of Ž 3. equals Ž x. BŽ x., where Ž.Ž 2. BŽ x. 2 as x. The left-had sde of 3 s 2 x Ž x. Ý x Ž x. CŽ x., where C x as x. It follows that f 2 Ž2,. Ž 3. holds for x suffcetly close to.
BRUNN MINKOWSKI INEQUALITY 5 Whe 2, 2 becomes 2 V Ž K, L. VŽ K L. 2. I ths case Ž. 7 mples that VŽ K L. 2 2V Ž K, L., so the equalty caot hold whe V Ž K, L. s large, o matter how large the costat s. The followg modfcato does ot requre the extra assumpto eeded Theorem 4.. THEOREM 4.2. Let K ad L be co ex bodes such that VŽ K. VŽ L.. The VŽ K, L. Ž VŽ K L. 2.. Proof. Followg the frst part of the proof of Theorem 4., ad aga Ž. lettg x V K, L, we see that t suffces to show that for x. Let Ž. The h 0 ad x Ž x. x 2, hž x. Ž x. x 2 Ž x.. 2 h Ž x. Ž x. x Ž. x 0, so h x 0 whe x, as requred. EXAMPLE 4.3. Let K ad L be rght sphercal cylders of rad r ad s, respectvely, ad of volume. The heghts of K ad L are Žr. ad Žs., respectvely. Sce K L s a cylder of radus r s ad heght equal to the sum of the heghts of K ad L, r s VŽ K L. Ž r s.. r s If 0, a smlar calculato gves VŽ K L. as a fucto of r, s, ad. The, usg Ž. 9, we fd that r Ž. s rs VŽ K, L.. rs
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